Here we present a detailed derivation of the Darboux–Halphen system starting from the Euclidean Bianchi IX metric with \({\rm SO}(3)\)  symmetry with an imposition of the constraints of self-duality at the level of Riemann curvature. The constraint of anti-self-duality yields an anti-instanton, a solution with negative instanton number and we present this solution together by using \(\pm \operatorname{sign}\) .
We follow the steps of [1]} and [2]}, see also [3]}.
